U.S. patent application number 14/285581 was filed with the patent office on 2015-10-01 for compensated tri-axial propagation measurements.
This patent application is currently assigned to Schlumberger Technology Corporation. The applicant listed for this patent is Schlumberger Technology Corporation. Invention is credited to Mark Frey.
Application Number | 20150276967 14/285581 |
Document ID | / |
Family ID | 54190035 |
Filed Date | 2015-10-01 |
United States Patent
Application |
20150276967 |
Kind Code |
A1 |
Frey; Mark |
October 1, 2015 |
Compensated Tri-Axial Propagation Measurements
Abstract
A method for obtaining fully gain compensated propagation
measurements includes rotating an electromagnetic logging while
drilling tool in a subterranean wellbore. The tool includes first
and second transmitters and first and second receivers axially
spaced apart from one another in which each of the transmitters and
each of the receivers include an axial antenna and collocated first
and second transverse antennas. The first and second transverse
antennas in the first receiver are rotationally offset by a
predefined angle from the first and second transverse antennas in
the first transmitter. A plurality of electromagnetic voltage
measurements are acquired at the first and second receivers while
rotating and processed to compute harmonic coefficients. The
harmonic coefficients are mathematically rotated through at least
the predefined angle to obtain rotated coefficients. Ratios of
selected ones of the rotated coefficients are processed to compute
fully gain compensated measurement quantities.
Inventors: |
Frey; Mark; (Sugar Land,
TX) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Schlumberger Technology Corporation |
Sugar Land |
TX |
US |
|
|
Assignee: |
Schlumberger Technology
Corporation
Sugar Land
TX
|
Family ID: |
54190035 |
Appl. No.: |
14/285581 |
Filed: |
May 22, 2014 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
61972291 |
Mar 29, 2014 |
|
|
|
Current U.S.
Class: |
702/7 |
Current CPC
Class: |
G01V 3/38 20130101; G01V
3/30 20130101 |
International
Class: |
G01V 3/26 20060101
G01V003/26; G01V 3/38 20060101 G01V003/38 |
Claims
1. A method for making downhole electromagnetic logging while
drilling measurements, the method comprising (a) rotating an
electromagnetic logging while drilling tool in a subterranean
wellbore, the logging tool including first and second transmitters
and first and second receivers axially spaced apart from one
another, each of the first and second transmitters and first and
second receivers including an axial antenna and collocated first
and second transverse antennas, the first and second transverse
antennas in the first receiver being rotationally offset by a
predefined angle from the first and second transverse antennas in
the first transmitter; (b) acquiring a plurality of electromagnetic
voltage measurements from the first and second receivers while
rotating in (a); (c) processing the voltage measurements acquired
in (b) to compute harmonic coefficients; (d) mathematically
rotating the harmonic coefficients through at least the predefined
angle to obtain rotated coefficients; and (e) processing ratios of
selected ones of the rotated coefficients to compute gain
compensated measurement quantities.
2. The method of claim 1, wherein the processing in (e) is
performed by a downhole processor.
3. The method of claim 2, further comprising: (f) transmitting the
gain compensated measurement quantities to a surface location; and
(g) causing a surface computer to invert the gain compensated
measurement quantities to obtain one or more properties of a
subterranean formation.
4. The method of claim 1, further comprising: (f) processing the
gain compensated measurement quantities to compute compensated
phase shift and attenuation quantities.
5. The method of claim 1, wherein: the first and second transverse
antennas in the second receiver are rotationally offset by the
predefined angle from the first and second transverse antennas in
the second transmitter; and the first and second transverse
antennas in the second transmitter are rotationally offset by an
arbitrary angle from the first and second transverse antennas in
the first transmitter.
6. The method of claim 1, wherein the electromagnetic voltage
measurements acquired in (b) comprise a three-dimensional matrix of
voltage measurements.
7. The method of claim 1, wherein the harmonic coefficients
computed in (c) comprise DC, first harmonic sine, first harmonic
cosine, second harmonic sine, and second harmonic cosine
coefficients.
8. The method of claim 1, wherein the gain compensated measurement
quantities computed in (e) comprise at least one measurement
proportional to an xx coupling, a yy coupling, an xx coupling plus
a yy coupling, or an xx coupling minus a yy coupling.
9. The method of claim 1, wherein the gain compensated measurement
quantities computed in (e) comprise at least one measurement
proportional to an xy coupling, a yx coupling, an xy coupling minus
a yx coupling, or an xy coupling plus a yx coupling.
10. The method of claim 1, wherein the gain compensated measurement
quantities computed in (e) comprise at least one measurement
proportional to a zx coupling, an xz coupling, a zy coupling, or a
yz coupling.
11. The method of claim 1, wherein the gain compensated measurement
quantities computed in (e) comprise at least one measurement
sensitive to a product of an xz coupling and a zx coupling or a
product of a yz coupling and a zy coupling.
12. The method of claim 1, wherein the gain compensated measurement
quantities computed in (e) comprise (i) at least one measurement
proportional to an xx coupling or a yy coupling, (ii) at least one
measurement proportional to an xy coupling or yx coupling, (iii) at
least one measurement proportional to a zx coupling, an xz
coupling, a zy coupling, or a yz coupling, and (iv) a measurement
proportional to an zz coupling.
13. The method of claim 1, further comprising: (f) processing the
compensated measurement quantities computed in (e) to compute
compensated symmetrized and anti-symmetrized measurement
quantities.
14. The method of claim 1, wherein (c) further comprises: (i)
processing the voltage measurements acquired in (b) to compute
harmonic coefficients, and (ii) processing selected ones of the
harmonic coefficients computed in (i) to obtain transmitter and
receiver gain ratio matrices, and (iii) applying the gain ratio
matrices to the harmonic coefficients.
15. The method of claim 1, wherein (e) further comprises: (i)
processing combinations of the rotated coefficients to obtain
rotated combinations and (ii) processing ratios of selected ones of
the rotated combinations to obtain the gain compensated measurement
quantities.
16. A method for making downhole electromagnetic logging while
drilling measurements, the method comprising (a) rotating an
electromagnetic logging while drilling tool in a subterranean
wellbore, the logging tool including first and second transmitters
and first and second receivers axially spaced apart from one
another, each of the first and second transmitters and first and
second receivers including an axial antenna and collocated first
and second transverse antennas, the first and second transverse
antennas in the first receiver being rotationally offset by a
predefined angle from the first and second transverse antennas in
the first transmitter; and (b) acquiring a plurality of
electromagnetic voltage measurements from the first and second
receivers while rotating in (a); (c) processing the voltage
measurements acquired in (b) to compute harmonic coefficients; (d)
processing selected ones of the harmonic coefficients computed in
(c) to obtain transmitter and receiver gain matrices; (e) applying
the gain matrices to the harmonic coefficients; (f) mathematically
rotating the harmonic coefficients through at least the predefined
angle to obtain rotated coefficients; (g) processing combinations
of the rotated coefficients to obtain rotated combinations; and (h)
processing ratios of selected ones of the rotated combinations to
obtain gain compensated measurement quantities.
17. The method of claim 16, wherein the processing in (e) is
performed by a downhole processor and the method further comprises:
(i) transmitting the gain compensated measurement quantities to a
surface location; and (j) causing a surface computer to invert the
gain compensated measurement quantities to obtain one or more
properties of a subterranean formation.
18. The method of claim 16, further comprising: (i) processing the
gain compensated measurement quantities to compute compensated
phase shift and attenuation quantities.
19. The method of claim 16, wherein the electromagnetic voltage
measurements acquired in (b) comprise a three-dimensional matrix of
voltage measurements.
20. The method of claim 16, wherein the gain compensated
measurement quantities computed in (h) comprise at least one
measurement proportional to an xx coupling, a yy coupling, or an xx
coupling minus a yy coupling.
21. The method of claim 16, wherein the gain compensated
measurement quantities computed in (h) comprise at least one
measurement proportional to an xy coupling, a yx coupling, an xy
coupling minus a yx coupling, or an xy coupling plus a yx
coupling.
22. The method of claim 16, wherein the gain compensated
measurement quantities computed in (h) comprise at least one
measurement proportional to a zx coupling, an xz coupling, a zy
coupling, or a yz coupling.
23. The method of claim 16, wherein the gain compensated
measurement quantities computed in (h) comprise at least one
measurement sensitive to a product of an xz coupling and a zx
coupling or a product of a yz coupling and a zy coupling.
24. The method of claim 16, wherein the gain compensated
measurement quantities computed in (h) comprise (i) at least one
measurement proportional to an xx coupling or a yy coupling, (ii)
at least one measurement proportional to an xy coupling or yx
coupling, (iii) at least one measurement proportional to a zx
coupling, an xz coupling, a zy coupling, or a yz coupling, and (iv)
a measurement proportional to an zz coupling.
25. The method of claim 16, further comprising: (i) processing the
compensated measurement quantities computed in (h) to compute
compensated symmetrized and anti-symmetrized measurement
quantities.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application claims the benefit of U.S. Provisional
Application Ser. No. 61/972,291 entitled Compensated Deep
Propagation Tensor Measurement with Orthogonal Antennas, filed Mar.
29, 2014.
FIELD OF THE INVENTION
[0002] Disclosed embodiments relate generally to downhole
electromagnetic logging and more particularly to a method for
making fully gain compensated tri-axial propagation measurements,
such as phase shift and attenuation measurements, using orthogonal
antennas.
BACKGROUND INFORMATION
[0003] The use of electromagnetic measurements in prior art
downhole applications, such as logging while drilling (LWD) and
wireline logging applications is well known. Such techniques may be
utilized to determine a subterranean formation resistivity, which,
along with formation porosity measurements, is often used to
indicate the presence of hydrocarbons in the formation. Moreover,
azimuthally sensitive directional resistivity measurements are
commonly employed e.g., in pay-zone steering applications, to
provide information upon which steering decisions may be made.
[0004] Downhole electromagnetic measurements are commonly inverted
at the surface using a formation model to obtain various formation
parameters, for example, including vertical resistivity, horizontal
resistivity, distance to a remote bed, resistivity of the remote
bed, dip angle, and the like. One challenge in utilizing
directional electromagnetic resistivity measurements, is obtaining
a sufficient quantity of data to perform a reliable inversion. The
actual formation structure is frequently significantly more complex
than the formation models used in the inversion. The use of a
three-dimensional matrix of propagation measurements may enable a
full three-dimensional measurement of the formation properties to
be obtained as well as improve formation imaging and
electromagnetic look ahead measurements. However, there are no
known methods for providing a fully gain compensated tri-axial
propagation measurement.
SUMMARY
[0005] A method for obtaining fully gain compensated propagation
measurements is disclosed. The method includes rotating an
electromagnetic logging while drilling tool in a subterranean
wellbore. The tool includes first and second transmitters and first
and second receivers axially spaced apart from one another in which
each of the first and second transmitters and first and second
receivers include an axial antenna and collocated first and second
transverse antennas. The first and second transverse antennas in
the first receiver are rotationally offset by a predefined angle
from the first and second transverse antennas in the first
transmitter. A plurality of electromagnetic voltage measurements
are acquired using the first and second receivers while rotating.
The measurements are processed to compute harmonic coefficients
which are in turn mathematically rotated through at least the
predefined angle to obtain rotated coefficients. Ratios of selected
ones of the rotated coefficients are processed to compute fully
gain compensated measurement quantities.
[0006] The disclosed embodiments may provide various technical
advantages. For example, the disclosed methodology provides a
method for obtaining a gain compensated three-dimensional matrix of
measurements using orthogonal antennas. The acquired measurements
are fully gain compensated and independent of antenna tilt angle
variation. Moreover, the disclosed method and apparatus tends to be
insensitive to bending and alignment angle errors.
[0007] This summary is provided to introduce a selection of
concepts that are further described below in the detailed
description. This summary is not intended to identify key or
essential features of the claimed subject matter, nor is it
intended to be used as an aid in limiting the scope of the claimed
subject matter.
BRIEF DESCRIPTION OF THE DRAWINGS
[0008] For a more complete understanding of the disclosed subject
matter, and advantages thereof, reference is now made to the
following descriptions taken in conjunction with the accompanying
drawings, in which:
[0009] FIG. 1 depicts one example of a drilling rig on which the
disclosed electromagnetic logging tools and methods may be
utilized.
[0010] FIG. 2A depicts one example of the deep reading
electromagnetic logging tool shown on FIG. 1.
[0011] FIG. 2B schematically depicts a deep reading electromagnetic
logging tool including collocated triaxial transmitters and
receivers.
[0012] FIG. 2C schematically depicts another deep reading
electromagnetic logging tool embodiment including transmitters and
receivers having axial and orthogonal antennas.
[0013] FIG. 3A depicts a flow chart of a disclosed method
embodiment.
[0014] FIG. 3B depicts a flow chart of another disclosed method
embodiment.
[0015] FIGS. 4A and 4B depict phase shift and attenuation versus
depth for a compensated zz measurement at frequencies of 2, 6, and
12 kHz.
[0016] FIGS. 5A and 5B depict phase shift and attenuation versus
depth for a compensated xx measurement at frequencies of 2, 6, and
12 kHz.
[0017] FIG. 6 depicts a three layer formation model used to
evaluate the directional response of disclosed symmetrized and
anti-symmetrized measurements.
[0018] FIGS. 7A and 7B depict symmetrized and anti-symmetrized
phase shift and attenuation versus total vertical depth at 30
degrees relative dip.
[0019] FIGS. 8A and 8B depict symmetrized and anti-symmetrized
phase shift and attenuation versus total vertical depth at 70
degrees relative dip.
DETAILED DESCRIPTION
[0020] FIG. 1 depicts an example drilling rig 10 suitable for
employing various method embodiments disclosed herein. A
semisubmersible drilling platform 12 is positioned over an oil or
gas formation (not shown) disposed below the sea floor 16. A subsea
conduit 18 extends from deck 20 of platform 12 to a wellhead
installation 22. The platform may include a derrick and a hoisting
apparatus for raising and lowering a drill string 30, which, as
shown, extends into borehole 40 and includes a drill bit 32
deployed at the lower end of a bottom hole assembly (BHA) that
further includes a deep reading electromagnetic measurement tool 50
configured to make tri-axial electromagnetic logging measurements.
As described in more detail below the deep reading electromagnetic
measurement tool 50 may include multiple orthogonal antennas
deployed on at least first and second axially spaced subs.
[0021] It will be understood that the deployment illustrated on
FIG. 1 is merely an example. Drill string 30 may include
substantially any suitable downhole tool components, for example,
including a steering tool such as a rotary steerable tool, a
downhole telemetry system, and one or more MWD or LWD tools
including various sensors for sensing downhole characteristics of
the borehole and the surrounding formation. The disclosed
embodiments are by no means limited to any particular drill string
configuration.
[0022] It will be further understood that the disclosed embodiments
are not limited to use with a semisubmersible platform 12 as
illustrated on FIG. 1. The disclosed embodiments are equally well
suited for use with either onshore or offshore subterranean
operations.
[0023] FIGS. 2A, 2B, and 2C (collectively FIG. 2) depict
electromagnetic measurement tool embodiments 50 and 50'. FIG. 2A
depicts one example of an electromagnetic measurement tool 50. In
the depicted embodiment measurement tool 50 includes a first
transmitter 52 and a first receiver 53 deployed on a first sub 51
and a second transmitter 57 and a second receiver 58 deployed on a
second sub 56. The first and second subs 51 and 56 may be axially
spaced apart substantially any suitable distance to achieve a
desired measurement depth. While not shown, other BHA tools may
deployed between subs 51 and 56. As described in more detail below
each of the transmitters 52 and 57 and receivers 53 and 58 includes
three tri-axial antennas (an axial antenna and first and second
transverse antennas that are orthogonal to one another in this
particular embodiment). As is known to those of ordinary skill in
the art, an axial antenna is one whose moment is substantially
parallel with the longitudinal axis of the tool. Axial antennas are
commonly wound about the circumference of the logging tool such
that the plane of the antenna is substantially orthogonal to the
tool axis. A transverse antenna is one whose moment is
substantially perpendicular to the longitudinal axis of the tool. A
transverse antenna may include, for example, a saddle coil (e.g.,
as disclosed in U.S. Patent Publications 2011/0074427 and
2011/0238312 each of which is incorporated by reference
herein).
[0024] FIG. 2B depicts the antenna moments for transmitters 52 and
57 and receivers 53 and 58 on electromagnetic measurement tool 50.
Transmitter 52 (T1) includes three collocated tri-axial antennas
having mutually orthogonal moments T1x, T1y, and T1z aligned with
the x-, y-, and z-directions. Receiver 53 (R1) also includes three
collocated tri-axial antennas having mutually orthogonal moments
R1x, R1y, and R1z. Moment R1z is aligned with T1z (and the z-axis)
while moments R1x and R1y are rotationally offset from T1x and T1y
by an offset angle .alpha.=45 degrees. As depicted sub 56 is
rotationally offset (about the axis of the drill string, the
z-axis) with respect to sub 51 by a misalignment angle .gamma. such
that transmitter T2 and receiver R2 are rotationally offset with
respect transmitter T1 and receiver R1. Transmitter 57 (T2)
includes collocated tri-axial antennas having three mutually
orthogonal moments T2x, T2y, and T2z. Moment T2z is aligned with
the z-axis while moments T2x and T2y are rotationally offset from
T1x and T1y by the angle .gamma. (as depicted at 55). Receiver 58
(R2) also includes collocated tri-axial antennas having mutually
orthogonal moments R2x, R2y, and R2z. Moment R2z is aligned with
T2z while moments R2x and R2y are rotationally offset from T2x and
T2y by .alpha.=45 degrees and from T1x and T1y by .alpha.+.gamma.
degrees. The disclosed embodiment tends to be advantageous in that
it ensures that the reception of a non-zero signal at each of the
x- and y-axis receivers when the x- and y-axis transmitters are
fired.
[0025] It will be understood that the offset angle .alpha. is not
necessarily 45 degrees as depicted on FIG. 2B, but may be
substantially any non-zero, non-ninety degree angle. An offset
angle .alpha. in a range from about 30 to about 60 degrees is
generally preferred, although by no means required. It will also be
understood that the misalignment angle .gamma. is the result of a
rotational misalignment between subs 51 and 56 during make-up of
the drill string. As such the misalignment angle .gamma. may have
substantially any value.
[0026] FIG. 2C depicts an alternative (and more general)
electromagnetic measurement tool embodiment 50'. Measurement tool
50' is similar to measurement tool 50 (FIGS. 2A and 2B) in that it
includes a first tri-axial transmitter T1' and a first tri-axial
receiver R1' deployed on a first sub 51' and a second tri-axial
transmitter T2' and a second tri-axial receiver R2' deployed on a
second sub 56'. Measurement tool 50' differs from measurement tool
50 in that the z-axis transmitter antennas T1z' and T2z' and the
z-axis receiver antennas R1z' and R2z' are not collocated with the
corresponding x- and y-axis transmitter and receiver antennas, but
are axially offset therefrom. In general, the x- and y-axis
transmitter and receiver antennas are collocated with one another
while collocation of any one or more of the corresponding z-axis
antennas is optional.
[0027] FIG. 3A depicts a flow chart of one disclosed method
embodiment 100. An electromagnetic measurement tool (e.g.,
measurement tool 50 or 50' on FIG. 2) is rotated in a subterranean
wellbore at 110. Electromagnetic measurements are acquired at 120
while the tool is rotating and processed to obtain harmonic voltage
coefficients. The harmonic voltage coefficients are rotated
mathematically at 130 to simulate rotation of the x and y antennas
in the R1 and R2 receivers and the T2 transmitter such that they
are rotationally aligned with the x and y antennas in the T1
transmitter. Such rotation removes the effect of the offset angle
.alpha. and misalignment angle .gamma. on the measurements. Ratios
of selected ones of the rotated voltage coefficients may then be
processed to obtain gain compensated measurement quantities at
140.
[0028] FIG. 3B depicts a flow chart of an optional, more detailed
embodiment 100' of the method 100 depicted on FIG. 3A. An
electromagnetic measurement tool is rotated in a subterranean
wellbore at 110. A three dimensional matrix of electromagnetic
voltage measurements is acquired at 120 while rotating. The
acquired measurements are processed at 122 to obtain harmonic
voltage coefficients (e.g., the DC, first harmonic cosine, first
harmonic sine, second harmonic cosine, and second harmonic sine
coefficients). At 124, selected ones of the harmonic voltage
coefficients are processed to obtain transmitter and receiver gain
ratio matrices. The harmonic voltage coefficients may be selected,
for example, so that the gain ratio matrices replace the y
transmitter and y receiver gains with x transmitter and x receiver
gains when applied to the harmonic voltage coefficients. These gain
ratio matrices may then be applied to the harmonic voltage
coefficients at 126. The harmonic voltage coefficients (with the
applied gain ratio matrices) are rotated mathematically at 130 as
described above with respect to FIG. 3A. At 142, selected
combinations of the rotated measurements obtained in 130 may be
processed to obtain coefficient combinations and thereby increase
the signal to noise ratio of the measurements. Selected ratios of
these coefficient combinations may then be further processed at 144
to obtain gain compensated measurement quantities.
[0029] As is known to those of ordinary skill in the art, a time
varying electric current (an alternating current) in a transmitting
antenna produces a corresponding time varying magnetic field in the
local environment (e.g., the tool collar and the formation). The
magnetic field in turn induces electrical currents (eddy currents)
in the conductive formation. These eddy currents further produce
secondary magnetic fields which may produce a voltage response in a
receiving antenna. The measured voltage in the receiving antennas
can be processed, as is known to those of ordinary skill in the
art, to obtain one or more properties of the formation.
[0030] In general the earth is anisotropic such that its electrical
properties may be expressed as a three-dimensional tensor which
contains information on formation resistivity anisotropy, dip, bed
boundaries and other aspects of formation geometry. It will be
understood by those of ordinary skill in the art that the mutual
couplings between the tri-axial transmitter antennas and the
tri-axial receiver antennas depicted on FIGS. 2B and 2C form a
three-dimensional matrix and thus may have sensitivity to a full
three-dimensional formation impedance tensor. For example, a
three-dimensional matrix of measured voltages V may be expressed as
follows:
V ij = [ V ijxx V ijxy V ijxz V ijyx V ijyy V ijyz V ijzx V ijzy V
ijzz ] = I i Z ij = [ I ix 0 0 0 I iy 0 0 0 I iz ] [ Z ijxx Z ijxy
Z ijxz Z ijyx Z ijyy Z ijyz Z ijzx Z ijzy Z ijzz ] ( 1 )
##EQU00001##
[0031] where V.sub.ij represent the three-dimensional matrix of
measured voltages, with i indicating the corresponding transmitter
triad (e.g., T1 or T2) and j indicating the corresponding receiver
triad (e.g., R1 or R2), I.sub.i represent the transmitter currents,
and Z.sub.ij represent the transfer impedances which depend on the
electrical and magnetic properties of the environment surrounding
the antenna pair in addition to the frequency, geometry, and
spacing of the antennas. The third and fourth subscripts indicate
the axial orientation of the transmitter and receiver antennas. For
example, V.sub.12xy represents a voltage measurement on the y-axis
antenna of receiver R2 from a firing of the x-axis antenna of
transmitter T1.
[0032] When bending of the measurement tool is negligible (e.g.,
less than about 10 degrees), the measured voltages may be modeled
mathematically, for example, as follows:
V.sub.ij=G.sub.Tim.sub.Ti.sup.tR.sub..theta..sup.tZ.sub.ijR.sub..theta.m-
.sub.RjG.sub.Rj (2)
[0033] where Z.sub.ij are matrices representing the triaxial tensor
couplings (impedances) between the locations of transmitter i and
receiver j, G.sub.Ti and G.sub.Rj are diagonal matrices
representing the transmitter and receiver gains, R.sub..theta.
represents the rotation matrix about the z-axis through angle
.theta., m.sub.Ti and m.sub.Rj represent the matrices of the
direction cosines for the transmitter and receiver moments at
.theta.=0, and the superscript t represents the transpose of the
corresponding matrix. The matrices in Equation 2 may be given, for
example, as follows:
Z ij = [ Z ijxx Z ijxy Z ijxz Z ijyx Z ijyy Z ijyz Z ijzx Z ijzy Z
ijzz ] ( 3 ) G Ti = [ g Tix 0 0 0 g Tiy 0 0 0 g Tiz ] ( 4 ) G Rj =
[ g Rjx 0 0 0 g Rjy 0 0 0 g Rjz ] ( 5 ) R .theta. = [ cos ( .theta.
) - sin ( .theta. ) 0 sin ( .theta. ) cos ( .theta. ) 0 0 0 1 ] ( 6
) ##EQU00002##
[0034] Using the T1x antenna direction as a reference direction,
the matrices of the direction cosines of the transmitter and
receiver moments may be given, for example, as follows:
m.sub.T1=I
m.sub.R1=R.sub..alpha.
m.sub.R2=R.sub..gamma.R.sub..alpha.
m.sub.T2=R.sub..gamma. (7)
[0035] where I represents the identity matrix, R.sub..alpha.
represents the rotation matrix about the z-axis through the angle
.alpha., and R.sub..gamma. represents the rotation matrix about the
z-axis through the angle .gamma..
[0036] Substituting Equation 7 into Equation 2 yields the following
mathematical expressions:
V.sub.11=G.sub.T1(R.sub..theta..sup.tZ.sub.11R.sub..theta.)R.sub..alpha.-
G.sub.R1
V.sub.12=G.sub.T1(R.sub..theta..sup.tZ.sub.12R.sub..theta.)R.sub..gamma.-
R.sub..alpha.G.sub.R2
V.sub.21=G.sub.T2R.sub..gamma..sup.t(R.sub..theta..sup.tZ.sub.21R.sub..t-
heta.)R.sub..alpha.G.sub.R1
V.sub.22=G.sub.T2R.sub..gamma..sup.t(R.sub..theta..sup.tZ.sub.22R.sub..t-
heta.)R.sub..gamma.R.sub..alpha.G.sub.R2 (8)
[0037] The rotated tensor couplings (shown in the parentheses in
Equation 8) may be expressed mathematically in harmonic form, for
example, as follows:
R.sub..theta..sup.tZ.sub.ijR.sub..theta.=Z.sub.DC.sub.--.sub.ij+Z.sub.FH-
C.sub.--.sub.ij cos(.theta.)+Z.sub.FHS.sub.--.sub.ij
sin(.theta.)+Z.sub.SHC.sub.--.sub.ij
cos(2.theta.)+Z.sub.SHS.sub.--.sub.ij sin(2.theta.) (9)
[0038] where Z.sub.DC.sub.--.sub.ij represents a DC (average)
coupling coefficient, and Z.sub.FHC.sub.--.sub.ij and
Z.sub.FHS.sub.--.sub.ij represent first order harmonic cosine and
first order harmonic sine coefficients (referred to herein as first
harmonic cosine and first harmonic sine coefficients), and
Z.sub.SHC.sub.--.sub.ij and Z.sub.SHS.sub.--.sub.ij represent
second order harmonic cosine and second order harmonic sine
coefficients (referred to herein as second harmonic cosine and
second harmonic sine coefficients) of the couplings. These
coefficients are shown below:
Z DC _ ij = [ Z ijxx + Z ijyy 2 ( Z ijxy - Z ijyx ) 2 0 - ( Z ijxy
- Z ijyx ) 2 Z ijxx + Z ijyy 2 0 0 0 Z ijzz ] Z FHC _ ij = [ 0 0 Z
ijxz 0 0 Z ijyz Z ijzx Z ijzy 0 ] Z FHS _ ij = [ 0 0 Z ijyz 0 0 - Z
ijxz Z ijzy - Z ijzx 0 ] Z SHC _ ij = [ Z ijxx - Z ijyy 2 ( Z ijxy
+ Z ijyx ) 2 0 ( Z ijxy + Z ijyx ) 2 - ( Z ijxx + Z ijyy ) 2 0 0 0
0 ] Z SHS ij = [ ( Z ijxy + Z ijyx ) 2 - ( Z ijxx - Z ijyy ) 2 0 -
( Z ijxx - Z ijyy ) 2 - ( Z ijxy + Z ijyx ) 2 0 0 0 0 ] ( 10 )
##EQU00003##
[0039] As stated above, the receiver antenna voltages are measured
at 120 while the tool rotates at 100 (FIGS. 3A and 3B). Following
the form of Equation 9, the measured voltages may be expressed
mathematically in terms of their harmonic voltage coefficients, for
example, as follows thereby enabling the harmonic coefficients to
be obtained (e.g., at 122 in FIG. 3B):
V.sub.ij=V.sub.DC.sub.--.sub.ij+V.sub.FHC.sub.--.sub.ij
cos(.theta.)+V.sub.FHS.sub.--.sub.ij
sin(.theta.)+V.sub.SHC.sub.--.sub.ij
cos(2.theta.)+V.sub.SHS.sub.--.sub.ij sin(2.theta.) (11)
[0040] Following Equation 2, the DC, first harmonic, and second
harmonic voltage coefficients may be modeled, for example, as
follows:
V.sub.DC.sub.--.sub.ij=G.sub.Tim.sub.Ti.sup.tZ.sub.DC.sub.--.sub.ijm.sub-
.RjG.sub.Rj
V.sub.FHC.sub.--.sub.ij=G.sub.Tim.sub.Ti.sup.tZ.sub.FHC.sub.--.sub.ijm.s-
ub.RjG.sub.Rj
V.sub.FHS.sub.--.sub.ij=G.sub.Tim.sub.Ti.sup.tZ.sub.FHS.sub.--.sub.ijm.s-
ub.RjG.sub.Rj
V.sub.SHC.sub.--.sub.ij=G.sub.Tim.sub.Ti.sup.tZ.sub.SHC.sub.--.sub.ijm.s-
ub.RjG.sub.Rj
V.sub.SHS.sub.--.sub.ij=G.sub.Tim.sub.Ti.sup.tZ.sub.SHS.sub.--.sub.ijm.s-
ub.RjG.sub.Rj (12)
[0041] In one disclosed embodiment gain compensation may be
accomplished by obtaining ratios between the x and y and receiver
gains and the x and y transmitter gains (e.g., at 124 in FIG. 3B).
The DC voltage measurements at receiver R1 upon firing transmitter
T1 may be expressed as follows:
V DC _ 11 = [ V DC _ 11 xx V DC _ 11 xy V DC _ 11 xz V DC _ 11 yx V
DC _ 11 yy V DC _ 11 yz V DC _ 11 zx V DC _ 11 zy V DC _ 11 zz ] (
13 ) ##EQU00004##
[0042] From Equations 10 and 12, the measured DC voltages
V.sub.DC.sub.--.sub.11 may be expressed as a function of the
couplings (impedances), gains, and the angle .alpha., for example,
as follows:
[ g T 1 x g R 1 x [ ( Z 11 xx + Z 11 yy ) 2 cos ( .alpha. ) + ( Z
11 xy - Z 11 yx ) 2 sin ( .alpha. ) ] g T 1 x g R 1 y [ ( Z 11 xy -
Z 11 yx ) 2 cos ( .alpha. ) - ( Z 11 xx + Z 11 yy ) 2 sin ( .alpha.
) ] 0 - g T 1 y g R 1 x [ ( Z 11 xy - Z 11 yx ) 2 sin ( .alpha. ) -
( Z 11 xx + Z 11 yy ) 2 cos ( .alpha. ) ] g T 1 y g R 1 y [ ( Z 11
xx + Z 11 yy ) 2 cos ( .alpha. ) + ( Z 11 xy - Z 11 yx ) 2 sin (
.alpha. ) ] 0 0 0 g T 1 z g T 1 z Z 11 zz ] ( 14 ) ##EQU00005##
[0043] Taking the ratio between the DC xx and yy voltage
measurements yields:
V DC _ 11 xx V DC _ 11 yy = g R 1 x g R 1 y g T 1 x g T 1 y ( 15 )
##EQU00006##
[0044] Likewise, taking the ratio between the DC voltage xy and yx
measurements yields:
V DC _ 11 xy V DC _ 11 yx = - g R 1 y g R 1 x g T 1 x g T 1 y ( 16
) ##EQU00007##
[0045] where g.sub.R1x and g.sub.R1y represent the gains of the x
and y antenna on receiver R1 and g.sub.T1x and g.sub.T1y represent
the gains of the x and y antenna on transmitter T1. Equations 15
and 16 may be combined to obtain measured quantities that are
equivalent to a gain ratio of the x and y receiver and a gain ratio
of the x and y transmitter, for example, as follows:
gR 1 = def - V DC 11 xx V DC 11 yy V DC 11 yx V DC 11 xy = g R 1 x
g R 1 y gT 1 = def - V DC_ 11 xx V DC_ 11 yy V DC_ 11 xy V DC_ 11
yx = g T1x g T 1 y ( 17 ) ##EQU00008##
[0046] Since the gain ratio formulas in Equation 17 involve taking
a square root, there may be a 180 degree phase ambiguity (i.e., a
sign ambiguity). As such, the gain ratios may not be arbitrary, but
should be controlled such that they are less than 180 degrees. For
un-tuned receiving antennas, the electronic and antenna gain/phase
mismatch (assuming the antenna wires are not flipped from one
receiver to another) may generally be controlled to within about 30
degrees (particularly at the lower frequencies used for deep
measurements). This is well within 180 degrees (even at elevated
temperatures where the mismatch may be at its greatest). For tuned
transmitting antennas, however, the phase may change signs (i.e.,
jump by 180 degrees) if the drift in the antenna tuning moves
across the tuning resonance. Such transmitter phase ratio ambiguity
(sign ambiguity) may be resolved, for example, using Equations 15
and 16 and the knowledge that the receiver gain/phase ratio is not
arbitrary, but limited to about 30 degrees (i.e. to enable the
determination of whether the transmitter phase difference is closer
to 0 or 180 degrees).
[0047] The x and y gain ratios defined in Equation 17 enable the
following gain ratio matrices to be defined (e.g., at 124 in FIG.
3B):
G R 1 _ ratio = def [ 1 0 0 0 gR 1 0 0 0 1 ] = [ 1 0 0 0 g R 1 x g
R 1 y 0 0 0 1 ] G T 1 _ ratio = def [ 1 0 0 0 gT 1 0 0 0 1 ] = [ 1
0 0 0 g T 1 x g T 1 y 0 0 0 1 ] ( 18 ) ##EQU00009##
[0048] where G.sub.R1.sub.--.sub.ratio represents the gain ratio
matrix for receiver R1 and G.sub.T1.sub.--.sub.ratio represents the
gain ratio matrix for transmitter T1. Similar gain ratio matrices
may be obtained for receiver R2 and transmitter T2.
[0049] Applying these gain ratios to the measured voltages (shown
in Equation 14) enables the y transmitter and y receiver gains to
be replaced by x transmitter and x receiver gains (e.g., at 126 in
FIG. 3B). The voltage measurements may then be rotated
mathematically (e.g., at 130 in FIG. 3B) to simulate rotation of
the x and y antennas in the R1 and R2 receivers and the T2
transmitter such that they are rotationally aligned with the x and
y antennas in the T1 transmitter. Such rotation removes the effect
of the offset angle .alpha. and misalignment angle .gamma. on the
measurements. For example, the DC voltages measured between T1 and
R1 may be back rotated by the measured alignment angle between T1
and R1 .alpha.m. The alignment angle may be measured using
substantially any technique, for example, including a physical
caliper measurement, and is referred to as .alpha.m to indicate
that it is a measured value. This process may be represented
mathematically, for example, as follows:
V DC _ 11 _ rot = def G T 1 _ ratio V DC _ 11 G R 1 _ ratio R
.alpha. m t = [ g T 1 x g R 1 x ( Z 11 xx + Z 11 yy ) 2 g T 1 x g R
1 x ( Z 11 xy - Z 11 yx ) 2 0 - g T 1 x g R 1 x ( Z 11 xy - Z 11 yx
) 2 g T 1 x g R 1 x ( Z 11 xx + Z 11 yy ) 2 0 0 0 g T 1 z g R 1 z Z
11 zz ] ( 19 ) ##EQU00010##
[0050] where V.sub.DC.sub.--.sub.11.sub.--.sub.rot represent the
rotated DC voltage coefficients. It will be understood that
rotation about the z-axis does not change the value of the DC
coefficient (see Equation 9) and that Equation 19 may be expressed
identically as:
V.sub.DC.sub.--.sub.11.sub.--.sub.rotG.sub.T1.sub.--.sub.ratioV.sub.DC.su-
b.--.sub.11G.sub.R1.sub.--.sub.ratio. Notwithstanding, in the
description that follows, the DC coefficients are shown to be
rotated to be consistent with the first harmonic and second
harmonic coefficients.
[0051] The first harmonic cosine coefficients may be similarly
rotated to obtain rotated first harmonic cosine coefficients, for
example, as follows:
V FHC _ 11 _ rot = def G T 1 _ ratio V FHC _ 11 G R 1 _ ratio R
.alpha. m t = [ 0 0 g T 1 x g R 1 z Z 11 xz 0 0 g T 1 x g R 1 z Z
11 yz g T 1 z g R 1 x Z 11 zx g T 1 z g R 1 x Z 11 zy 0 ] ( 20 )
##EQU00011##
[0052] where V.sub.FHC.sub.--.sub.11.sub.--.sub.rot represent the
rotated first harmonic cosine voltage coefficients. The first
harmonic cosine coefficients may be similarly rotated by .alpha.m
plus an additional 90 degree back rotation to obtain rotated first
harmonic sine coefficients, for example, as follows:
V FHS _ 11 _ rot = def R 90 G T 1 _ ratio V FHS _ 11 G R 1 _ ratio
R .alpha. m t R 90 t = [ 0 0 g T 1 x g R 1 z Z 11 xz 0 0 g T 1 x g
R 1 z Z 11 yz g T 1 z g R 1 x Z 11 zx g T 1 z g R 1 x Z 11 zy 0 ] (
21 ) ##EQU00012##
[0053] where V.sub.FHS.sub.--.sub.11.sub.--.sub.rot represent the
rotated first harmonic sine voltage coefficients. The second
harmonic cosine coefficients may be rotated similarly to the first
harmonic cosine coefficients to obtain rotated second harmonic
cosine coefficients, for example, as follows:
V SHC _ 11 _ rot = def G T 1 _ ratio V SHC _ 11 G R 1 _ ratio R
.alpha. m t = [ g T 1 x g R 1 x ( z 11 xx - z 11 yy ) 2 g T 1 x g R
1 x ( z 11 xy + z 11 yx ) 2 0 g T 1 x g R 1 x ( z 11 xy + z 11 yx )
2 - g T 1 x g R 1 x ( z 11 xx - z 11 yy ) 2 0 0 0 0 ] ( 22 )
##EQU00013##
[0054] where V.sub.SHC.sub.--.sub.11.sub.--.sub.rot represent the
rotated second harmonic cosine voltage coefficients. The second
harmonic cosine coefficients may be similarly rotated by .alpha.m
plus an additional 45 degree back rotation to obtain rotated second
harmonic sine coefficients, for example, as follows:
V SHS _ 11 _ rot = def R 45 G T 1 _ ratio V SHS _ 11 G R 1 _ ratio
R .alpha. m t R 45 t = [ g T 1 x g R 1 x ( z 11 xx - z 11 yy ) 2 g
T 1 x g R 1 x ( z 11 xy + z 11 yx ) 2 0 g T 1 x g R 1 x ( z 11 xy +
z 11 yx ) 2 - g T 1 x g R 1 x ( z 11 xx - z 11 yy ) 2 0 0 0 0 ] (
23 ) ##EQU00014##
[0055] where V.sub.SHS.sub.--.sub.11.sub.--.sub.rot represent the
rotated second harmonic sine voltage coefficients. The voltage
measurements for other transmitter receiver combinations may also
be similarly rotated. For example, the voltage measurements on
receiver R2 obtained upon firing transmitter T1 may be back rotated
by both .alpha.m and the measured alignment mismatch between the
first and second subs .gamma.m (as though receiver R2 were back
rotated with respect to transmitter T1 by .alpha.m and .gamma.m).
The misalignment angle between the subs may be measured using
substantially any technique. For example, the misalignment angle
may be taken to be the difference between magnetic toolface angles
measured at each of the subs, and is referred to as .gamma.m to
indicate that it is a measured value. The T1-R2 voltage
measurements may be given, for example, as follows:
V DC _ 12 _ rot = def G T 1 _ ratio V DC _ 12 G R 2 _ ratio R
.alpha. m t R .gamma. m t V FHC _ 12 _ rot = def G T 1 _ ratio V
FHC _ 12 G R 2 _ ratio R .alpha. m t R .gamma. m t V FHS _ 12 _ rot
= def R 90 G T 1 _ ratio V FHS _ 12 G R 2 _ ratio R .alpha. m t R
90 t R .gamma. m t V SHC _ 12 _ rot = det G T 1 _ ratio V SHC _ 12
G R 2 _ ratio R .alpha. m t R .gamma. m t V SHS _ 12 _ rot = def R
45 G T 1 _ ratio V SHS _ 12 G R 2 _ ratio R .alpha. m t R 45 t R
.gamma. m t ( 24 ) ##EQU00015##
[0056] The voltage measurements on receiver R1 obtained upon firing
transmitter T2 may also be rotated (in this case as though receiver
R1 were back rotated with respect to transmitter T1 by .alpha.m and
transmitter T2 were back rotated with respect to transmitter T1 by
.gamma.m).
V DC _ 21 _ rot = def R .gamma. m G T 2 _ ratio V DC _ 21 G R 1 _
ratio R .alpha. m t V FHC _ 21 _ rot = def R .gamma. m G T 2 _
ratio V FHC _ 21 G R 1 _ ratio R .alpha. m t V FHS _ 21 _ rot = def
R .gamma. m R 90 G T 2 _ ratio V FHS _ 21 G R 1 _ ratio R .alpha. m
t R 90 t V SHC _ 21 _ rot = det R .gamma. m G T 2 _ ratio V SHC _
21 G R 1 _ ratio R .alpha. m t V SHS _ 21 _ rot = def R .gamma. m R
45 G T 2 _ ratio V SHS _ 21 G R 1 _ ratio R .alpha. m t R 45 t ( 25
) ##EQU00016##
[0057] The voltage measurements on receiver R2 obtained upon firing
transmitter T2 may also be rotated (in this case as though receiver
R2 were back rotated with respect to transmitter T1 by .alpha.m and
.gamma.m and transmitter T2 were back rotated with respect to
transmitter T1 by .gamma.m).
V DC _ 22 _ rot = def R .gamma. m G T 2 _ ratio V DC _ 22 G R 2 _
ratio R .alpha. m t R .gamma. m t V FHC _ 22 _ rot = def R .gamma.
m G T 2 _ ratio V FHC _ 22 G R 2 _ ratio R .alpha. m t R .gamma. m
t V FHS _ 22 _ rot = def R .gamma. m R 90 G T 2 _ ratio V FHS _ 22
G R 2 _ ratio R .alpha. m t R 90 t R .gamma. m t V SHC _ 22 _ rot =
det R .gamma. m G T 2 _ ratio V SHC _ 22 G R 2 _ ratio R .alpha. m
t R .gamma. m t V SHS _ 22 _ rot = def R .gamma. m R 45 G T 2 _
ratio V SHS _ 22 G R 2 _ ratio R .alpha. m t R 45 t R .gamma. m t (
26 ) ##EQU00017##
[0058] The rotated voltage measurements presented in Equations
19-26 may be combined in various combinations to obtain a large
number of compensated measurements (e.g., at 130 in FIG. 3A).
Selected ones of these compensated measurements are presented
below. For example, compensated quantities RCXX and RCYY equivalent
to the xx and yy direct coupling impedances (also referred to
herein as the xx and yy couplings) may be obtained as follows:
RCXX = ( V DC _ 12 xx _ rot + V SCH _ 12 xx _ rot ) ( V DC _ 21 xx
_ rot + V SCH _ 21 xx _ rot ) ( V DC _ 11 xx _ rot + V SCH _ 11 xx
_ rot ) ( V DC _ 22 xx _ rot + V SCH _ 22 xx _ rot ) = z 12 xx z 21
xx z 11 xx z 11 xx ( 27 ) RCYY = ( V DC _ 12 yy _ rot + V SCH _ 12
yy _ rot ) ( V DC _ 21 yy _ rot + V SCH _ 21 yy _ rot ) ( V DC _ 11
yy _ rot + V SCH _ 11 yy _ rot ) ( V DC _ 22 yy _ rot + V SCH _ 22
yy _ rot ) = z 12 yy z 21 yy z 11 yy z 11 yy ( 28 )
##EQU00018##
[0059] Compensated quantities RCXY and RCYX equivalent to the xy
and yx cross coupling impedances (also referred to herein as the xy
and yx couplings) may be obtained, for example, as follows:
RCXYij = V SHS_ijxy _rot + V DC_ijxy _rot 2 V DC_ijxx _rot = z ijxy
( z ijxx + z ijyy ) ( 29 ) RCYXij = V SHS_ijxy _rot - V DC_ijxy
_rot 2 V DC_ijxx _rot = z ijyx ( z ijxx + z ijyy ) ( 30 )
##EQU00019##
[0060] Compensated quantities RCXZ and RCYZ which are related to
the xz and zx cross coupling impedances and the yz and zy cross
coupling impedances (also referred to herein as the xz, zx, yz, and
zy couplings) may be obtained, for example, as follows:
RCXZij = V FHC_ijzz _rot V FHC_ijzx _rot V DC_ijxx _rot V DC_ijzz
_rot = 2 z ijxz z ijzx ( z ijxx + z ijyy ) z ijzx ( 31 ) RCYZij = V
FHC_ijyz _rot + V FHC_ijzy _rot V DC_ijyy _rot + V DC_ijzz _rot = 2
z ijxz z ijzx ( z ijxx + z ijyy ) z ijzx ( 32 ) ##EQU00020##
[0061] For each transmitter receiver combination the above
described rotated voltage coefficients (Equations 19-26) may also
be combined to improve signal to noise ratio (e.g., at 142 in FIG.
3B). For example, the following combinations may be obtained:
XXplusYY ij = def V D C ijxx_rot + V D C ijyy_rot 2 = g Tix g Rjx (
z ijxx + z ijyy ) 2 ( 33 ) XYminusYX ij = def V D C ijxy_rot + V D
C ijyx_rot 2 = g Tix g Rjx ( z ijxy + z ijyx ) 2 ( 34 ) XXminusYY
ij = def V SHC ijxx_rot - V SHC ijyy_rot + V SHS ijxx_rot - V SHS
ijyy_rot 4 = g Tix g Rjx ( z ijxx - z ijyy ) 2 ( 35 ) XYplusYX ij =
def V SHC ijxy_rot + V SHC ijyx_rot + V SHS ijxy_rot + V SHS
ijyx_rot 4 = g Tix g Rjx ( z ijxy - z ijyx ) 2 ( 36 ) XZ ij = def V
FHC ijxz_rot + V FHS ijxz_rot 2 = g Tix g Rjz Z ijxz ( 37 ) YZ ij =
def V FHC ijyz_rot + V FHS ijyz_rot 2 = g Tix g Rjz Z ijyz ( 38 )
ZX ij = def V FHC ijzx_rot + V FHS ijzx_rot 2 = g Tiz g Rjx Z ijzx
( 39 ) ZY ij = def V FHC ijzy_rot + V FHS ijzy_rot 2 = g Tiz g Rjx
Z ijzy ( 40 ) ##EQU00021##
[0062] The measurement equivalent to the zz coupling does not
required rotation and may be expressed, for example, as
follows:
ZZ ij = def V D C ijzz = g Tiz g Rjz Z ijzz ( 41 ) ##EQU00022##
[0063] The combined measurements in Equations 33 through 41 may be
further combined to fully compensate the transmitter and receiver
gains (e.g., at 144 in FIG. 3B). A compensated measurement
equivalent to the sum of the xx and yy couplings may be given, for
example, as follows:
CXXplusYY = XXplusYY 12 XXplusYY 21 XXplusYY 11 XXplusYY 22 = ( z
12 xx + z 12 yy ) ( z 21 xx + z 21 yy ) ( z 11 xx + z 12 yy ) ( z
21 xx + z 21 yy ) ( 42 ) ##EQU00023##
[0064] where CXXplusYY represents a compensated measurement
equivalent to the xx+yy coupling and XXplusYY.sub.ij is defined in
Equation 33. A phase shift and attenuation for this quantity may be
computed, for example, as follows:
XXplusYY_CPS = 180 .pi. angle ( CXXplusYY ) XXplusYY_CAD = 20 log
10 ( CXXplusYY ) ( 43 ) ##EQU00024##
[0065] where XXplusYY_CPS and XXplusYY_CAD represent the
compensated phase shift and attenuation of the xx+yy coupling.
[0066] Compensated measurements equivalent to the xx and/or yy
couplings may be constructed by combining the xx+yy measurements
with the xx-yy measurements, for example, as follows:
CXX = ( XXplusYY 12 + XXminusYY 12 ) ( XXplusYY 21 + XXminusYY 21 )
( XXplusYY 11 + XXminusYY 11 ) ( XXplusYY 22 + XXminusYY 22 ) = z
12 xx z 21 xx z 11 xx z 22 xx ( 44 ) CYY = ( XXplusYY 12 -
XXminusYY 12 ) ( XXplusYY 21 - XXminusYY 21 ) ( XXplusYY 11 -
XXminusYY 11 ) ( XXplusYY 22 - XXminusYY 22 ) = z 12 yy z 21 yy z
11 yy z 22 yy ( 45 ) ##EQU00025##
[0067] where CXX and CYY represent compensated measurements
equivalent to the xx and yy couplings and XXplusYY.sub.ij and
XXminusYY.sub.ij are defined in Equations 33 and 35. Phase shift
and attenuation for these quantities may be computed as described
above with respect to Equation 43.
[0068] Compensated measurements sensitive to a sum of the xy and yx
couplings may be computed in a similar manner using the second
harmonic cosine and the DC coefficients, for example, as
follows:
CXYplusYX = XYplusYX 12 XYplusYX 21 XXplusYY 11 XXplusYY 22 = ( z
12 xy + z 12 yx ) ( z 21 xy + z 21 yx ) ( z 11 xx + z 11 yy ) ( z
22 xx + z 22 yy ) ( 46 ) ##EQU00026##
[0069] where CXYplusYX represents the compensated measurement and
XYplusYX.sub.ij is defined in Equation 36. A compensated
measurement sensitive to a difference between the xy and yx
couplings may similarly be computed.
CXYminusYX = XYminusYX 12 XYminusYX 21 XXplusYY 11 XXplusYY 22 = (
Z 12 xy - Z 12 yx ) ( Z 21 xy - Z 21 yx ) ( Z 11 xx + Z 11 yy ) ( Z
22 xx + Z 22 yy ) ( 47 ) ##EQU00027##
[0070] where CXYminusYX represents the compensated measurement and
XXplusYY.sub.ij and XYminusYX.sub.ij and are defined in Equations
33 and 34. A compensated measurement sensitive to a difference
between the xx and yy couplings may further be computed:
CXXminusYY = XXminusYY 12 XXminusYY 21 XXplusYY 11 XXplusYY 22 = (
Z 12 xx - Z 12 yy ) ( Z 21 xx - Z 21 yy ) ( Z 11 xx + Z 11 yy ) ( Z
22 xx + Z 22 yy ) ( 48 ) ##EQU00028##
[0071] where CXXminusYY represents the compensated quantity and
XXplusYY.sub.11 and XXminusYY.sub.12 are defined in Equations 33
and 35.
[0072] Since the quantities in Equations 46, 47, and 48 may be
equal to zero in simple formations, the phase shift and attenuation
may be computed by adding one to the compensated quantity, for
example, as follows:
XYplusYX_CPS = 180 .pi. angle ( 1 + CXYplusYX ) XYplusYX_CAD = 20
log 10 ( 1 + CXYplusYX ) ( 49 ) XXminusYY_CPS = 180 .pi. angle ( 1
+ CXXminusYY ) XXminusYY_CAD = 20 log 10 ( 1 + CXXminusYY ) ( 50 )
XYminusYX_CPS = 180 .pi. angle ( 1 + CXYminusYX ) XYminusYX_CAD =
20 log 10 ( 1 + CXYminusYX ) ( 51 ) ##EQU00029##
[0073] where CPS quantities represent a compensated phase shift and
CAD quantities represent a compensated attenuation.
[0074] Other compensated combinations of the couplings may be
computed from the ratios of the second harmonic to DC coefficients.
These couplings are similar to those described above, but yield
compensated measurements at different depths of investigation
(i.e., using a single transmitter and a single receiver). For
example,
CXXminusYYij = XXminusYY ij XXplusYY ij = ( Z ijxx - Z ijyy ) ( Z
ijxx + Z ijyy ) ( 52 ) CXYplusYXij = XYplusYXij XXplusYY ij = ( Z
ijxy + Z ijyx ) ( Z ijxx + Z ijyy ) ( 53 ) CXYminusYXij =
XYminusYXij XXplusYY ij = ( Z ijxy - Z ijyx ) ( Z ijxx + Z ijyy ) (
54 ) ##EQU00030##
[0075] where CXXminusYYij, CXYplusYXij, and CXYminusYXij represent
the compensated measurements at any transmitter i receiver j
combination and XXplusYY.sub.ij, XYminusYXij, XXminusYY.sub.ij, and
XYplusYXij are defined in Equations 33 through 36. Additionally,
compensated combinations may be computed from the second harmonic
coefficients that are sensitive to either the xy or yx couplings
For example,
CXYij = XYplusYXij + XYminusYXij XXplusYYij = 2 Z ijxy ( Z ijxx + Z
ijyy ) ( 55 ) CYXij = XYplusYXijxx - XYminusYXij XXplusYYij = 2 Z
ijyx ( Z ijxx + Z ijyy ) ( 56 ) ##EQU00031##
[0076] where CXYij and CYXij represent the compensated measurements
at any transmitter i receiver j combination and XXplusYY.sub.ij,
XYminusYXij, and XYplusYXij are defined in Equations 33, 34, and
36. Phase shift and attenuation for the quantities shown in
Equations 52 through 56 may be computed as described above with
respect to Equations 49-51.
[0077] Equations 42 through 56 disclose compensated measurements
sensitive to one or more of the xx and yy couplings and the xy and
yx couplings. The zz coupling may be compensated, for example, as
follows:
CZZ = ZZ 12 ZZ 21 ZZ 11 ZZ 22 = Z 12 zz Z 21 zz Z 11 zz Z 22 zz (
57 ) ##EQU00032##
[0078] where CZZ represents the compensated measurement and
ZZ.sub.ij are defined in Equation 41.
[0079] Gain compensated quantities sensitive to the xz, zx, yz, and
zy couplings may be computed, for example, as follows:
CXZZXij = ZX ij ZZ ij XZ ij XXplusYY ij = 2 Z ijzx Z ijxz Z ijzz (
Z ijxx + Z ijyy ) ( 58 ) CYZZYij = ZY ij ZZ ij YZ ij XXplusYY ij =
2 Z ijzy Z ijyz Z ijzz ( Z ijxx + Z ijyy ) ( 59 ) ##EQU00033##
[0080] where CXZZXij represents a compensated quantity sensitive to
the product of xz and zx impedances, CYZZYij represents a
compensated quantity sensitive to the product of yz and zy
impedances, and XXplusYY.sub.ij, XZ.sub.ij, YZ.sub.ij, ZX.sub.ij,
and ZZ.sub.ij are defined in Equations 33 and 37 through 41. It
will be understood that CXZZXij and CYZZYij may be used to provide
compensated measurements at different depths of investigation
(e.g., at shallow depths for T1-R1 and T2-R2 combinations and
larger depths for T1-R2 and T2-R1 combinations).
[0081] Equations 33 and 37 through 41 may also be used to provide
compensated quantities that have properties similar to the
symmetrized and anti-symmetrized quantities disclosed in U.S. Pat.
Nos. 6,969,994 and 7,536,261 which are fully incorporated by
reference herein. For example, the following compensated ratios may
be computed.
R zx = def ZX 12 ZZ 11 XZ 21 XXplusYY 22 R xz = def XZ 12 XXplusYY
11 ZX 21 ZZ 22 R 1 xz _ zx = def ZX 12 ZZ 12 XZ 12 XXplusYY 12 R 2
xz _ zx = def ZX 21 ZZ 21 XZ 21 XXplusYY 21 ( 60 ) ##EQU00034##
[0082] In Equation 60, R.sub.zx and R.sub.xz represent compensated
quantities that are proportional to the square of the zx and xz
couplings. Hence, compensated measurements proportional to the zx
and xz couplings may be obtained, for example, as follows: CZX=
{square root over (R.sub.zx)} and CXZ= {square root over
(R.sub.zx)}. Gain compensated measurements sensitive to the zy and
yz couplings may be obtained similarly (i.e., by computing R.sub.zy
and R.sub.yz).
[0083] The compensated symmetrized and anti-symmetrized
measurements may then be defined, for example, as follows:
Ac = def 2 R zx + R xz + scale ( R 1 xz _ zx + R 2 xz _ zx ) Sc =
def 2 R zx + R xz - scale ( R 1 xz _ zx + R 2 xz _ zx ) ( 61 )
##EQU00035##
[0084] where
scale = def CZZ CXX ##EQU00036##
and R.sub.zx, R.sub.xz, R1.sub.xz.sub.--.sub.zx, and
R2.sub.xz.sub.--.sub.zx are defined in Equation 60. As described
above with respect to Equation 17, taking the square root of a
quantity can introduce a sign (or phase) ambiguity. Even with
careful unwrapping of the phase in Equation 61, a symmetrized
directional measurement Sc may have the same sign whether an
approaching bed is above or below the measurement tool. The correct
sign may be selected, for example, via selecting the sign of the
phase angle and/or attenuation of the following relation:
TSD= {square root over (R.sub.zx)}- {square root over (R.sub.xz)}
(62)
[0085] Similarly the anti-symmetrized directional measurement Ac in
Equation 61 has the same sign whether the dip azimuth of the
anisotropy is less than 180 degrees or greater than 180 degrees.
This sign ambiguity may be resolved, for example, by taking the
sign of the phase angle and/or attenuation of the following
relation.
TAD= {square root over (R.sub.zx)}+ {square root over (R.sub.xz)}
(63)
[0086] The symmetrized and anti-symmetrized measurements may now be
re-defined, for example, as follows to eliminate the sign
ambiguity.
Sc = def 2 sign ( angle ( TSD ) ) R zx + R xz - scale ( R 1 xz _ zx
+ R 2 xz _ zx ) Ac = def 2 sign ( angle ( TAD ) ) R zx + R xz +
scale ( R 1 xz _ zx + R 2 xz _ zx ) ( 64 ) ##EQU00037##
[0087] Symmetrized directional phase shift and attenuation
measurements TDSP and TDSA may be defined, for example, as
follows:
TDSP = def 180 .pi. angle ( 1 + Sc ) TDSA = def 20 log 10 ( 1 + Sc
) ( 65 ) ##EQU00038##
[0088] Likewise, anti-symmetrized directional phase shift and
attenuation TDAP and TDAA measurements may also be defined, for
example, as follows:
TDAP = def 180 .pi. angle ( 1 + Ac ) TDAA = def 20 log 10 ( 1 + Ac
) ( 66 ) ##EQU00039##
[0089] The symmetrized and anti-symmetrized phase shift and
attenuation given in Equations 65 and 66 may alternatively and/or
additionally be modified to scale the phase shifts and
attenuations. For example, for a deep reading array having a large
spreading loss the phase shifts in particular tend to be small.
These values can be scaled by the spreading loss to scale them to
values similar to those computed for a conventional shallow array,
for example, as follows:
TDSP = def 180 .pi. angle ( 1 + Sc SL ) TDSA = def 20 log 10 ( 1 +
Sc SL ) and ( 67 ) TDAP = def 180 .pi. angle 1 ( 1 + Ac SL ) TDAA =
def 20 log 10 ( 1 + Ac SL ) ( 68 ) ##EQU00040##
[0090] where SL represents the spreading loss which is proportional
to the cube of the ratio of the distance from the transmitter to
the near receiver to the distance from the transmitter to the far
receiver.
[0091] The quantities TSD and TAD computed in Equations 62 and 63
may alternatively be used to compute symmetrized and
anti-symmetrized phase shift and attenuation, for example, as
follows:
TDSP = def 180 .pi. angle ( 1 + TSD ) TDSA = def 20 log 10 ( 1 +
TSD ) and ( 69 ) TDAP = def 180 .pi. angle ( 1 + TAD ) TDAA = def
20 log 10 ( 1 + TAD ) ( 70 ) ##EQU00041##
[0092] Moreover, the quantities computed in Equations 58, 59, and
60 may also be used to compute phase shift and attenuation values
using the methodology in Equations 69 and 70.
[0093] The disclosed embodiments are now described in further
detail with respect to the following non-limiting examples in FIGS.
4A, 4B, 5A, 5B, 6, 7A, 7B, 8A, and 8B. These examples are
analytical (mathematical) and were computed using software code
developed based on a point dipole model.
[0094] In the examples that follow (in FIGS. 4A though 5B), a tool
model configuration similar to that shown on FIG. 2B was used in
which receivers R1 and R2, and transmitter T2 were located 7, 63,
and 70 feet above transmitter T1. A two-layer formation model was
used in which the upper bed had a horizontal resistivity of 2 ohmm
and a vertical resistivity of 4 ohmm and the lower bed had zero
conductivity (i.e., infinite resistivity). Zero depth was defined
as the depth at which the transmitter T1 crossed the bed
boundary.
[0095] FIGS. 4A and 4B depict phase shift and attenuation versus
depth for a compensated zz measurement (from Equation 57) at
frequencies of 2, 6, and 12 kHz for the above described model. The
computed phase shifts (FIG. 4A) depended on the measurement
frequency and were constant with depth above the bed boundary (in
the first layer) and decreased to zero below the bed boundary. The
computed phase shift values were zero at depths greater than about
100 feet. The computed attenuations (FIG. 4B) also depended on the
measurement frequency and were constant with depth above the bed
boundary (in the first layer) and decreased to about 57.5 dB below
the bed boundary. The computed phase shift values were independent
of depth at depths greater than about 100 feet.
[0096] FIGS. 5A and 5B depict phase shift and attenuation versus
depth for a compensated xx measurement (from Equation 44) at
frequencies of 2, 6, and 12 kHz. The computed phase shifts (FIG.
5A) depended on the measurement frequency and were constant with
depth down to about 50 feet above the bed boundary. The computed
phase shift values were zero at depths greater than about 100 feet
(below the boundary). Upon approaching the boundary (from above),
the phase shift values underwent a perturbation in which they first
decreased, increased sharply to a maximum just below the boundary,
decreased to a minimum about 70 feet below the boundary before
rising modestly to zero degrees. The computed attenuations (FIG.
4B) also depended on the measurement frequency and were constant
with depth down to about 50 feet above the bed boundary. The
attenuation was constant and independent of frequency at depths
above about 100 feet. Upon approaching the boundary (from above),
the phase shift values underwent a perturbation in which they first
increased sharply to a maximum just below the boundary, decreased
sharply to a minimum at about 70 feet below the boundary before
rising to just over 57 dB.
[0097] FIG. 6 depicts a three layer formation model used to
evaluate the directional response of the compensated symmetrized
and anti-symmetrized measurements shown in Equation 58. The upper
layer has a horizontal resistivity of 2 ohmm and a vertical
resistivity of 5 ohmm. The middle layer has a horizontal and
vertical resistivities of 200 ohmm while the lower layer has a
horizontal resistivity of 5 ohmm and a vertical resistivity of 10
ohmm. The upper and lower boundaries of the middle layer were at
-125 and 125 feet, respectively. The electromagnetic tool was
inclined at a non-zero dip angle D. In the examples that follow (in
FIGS. 7A though 8B), a tool model configuration similar to that
shown on FIG. 2B was used in which receivers R1 and R2, and
transmitter T2 were located 7, 63, and 70 feet above transmitter
T1. Zero depth here was defined as the depth at which the mid-point
between the receivers crossed the midpoint of the middle layer.
[0098] FIGS. 7A-8B depict symmetrized (solid) and anti-symmetrized
(dashed) phase shift and attenuation versus total vertical depth at
30 degrees (FIGS. 7A and 7B) and 70 degrees (FIGS. 8A and 8B)
relative dip. The symmetrized phase shift and attenuation are zero
away from the bed boundary. Near the boundaries the symmetrized
measurement shows a strong response that is independent of
anisotropy and whose sign depends on whether the bed is being
approached from above or below. The anti-symmetrized phase shift
and attenuation respond to dip and anisotropy away from the bed
boundary. The anti-symmetrized measurement response to a boundary
is suppressed compared to the symmetrized.
[0099] It will be understood that the various methods disclosed
herein for obtaining fully gain compensated electromagnetic
measurement quantities may be implemented on a on a downhole
processor. By downhole processor it is meant an electronic
processor (e.g., a microprocessor or digital controller) deployed
in the drill string (e.g., in the electromagnetic logging tool or
elsewhere in the BHA). In such embodiments, the fully compensated
measurement quantities may be stored in downhole memory and/or
transmitted to the surface while drilling via known telemetry
techniques (e.g., mud pulse telemetry or wired drill pipe).
Alternatively, the harmonic fitting coefficients may transmitted
uphole and the compensated quantities may be computed at the
surface using a surface processor. Whether transmitted to the
surface or computed at the surface, the quantity may be utilized in
an inversion process (along with a formation model) to obtain
various formation parameters as described above.
[0100] Although compensated tri-axial propagation measurements have
been described in detail, it should be understood that various
changes, substitutions and alternations can be made herein without
departing from the spirit and scope of the disclosure as defined by
the appended claims.
* * * * *